ZHBGVX(l) ) ZHBGVX(l)
NAME
ZHBGVX - compute all the eigenvalues, and optionally, the eigenvectors of a complex gener-
alized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x
SYNOPSIS
SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL,
IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
PURPOSE
ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex gener-
alized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B
are assumed to be Hermitian and banded, and B is also positive definite. Eigenvalues and
eigenvectors can be selected by specifying either all eigenvalues, a range of values or a
range of indices for the desired eigenvalues.
ARGUMENTS
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input) CHARACTER*1
= 'A': all eigenvalues will be found;
= 'V': all eigenvalues in the half-open interval (VL,VU] will be found; = 'I': the
IL-th through IU-th eigenvalues will be found.
UPLO (input) CHARACTER*1
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
KA (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U', or the number of sub-
diagonals if UPLO = 'L'. KA >= 0.
KB (input) INTEGER
The number of superdiagonals of the matrix B if UPLO = 'U', or the number of sub-
diagonals if UPLO = 'L'. KB >= 0.
AB (input/output) COMPLEX*16 array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Hermitian band matrix A, stored in
the first ka+1 rows of the array. The j-th column of A is stored in the j-th col-
umn of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for
max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
On exit, the contents of AB are destroyed.
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KA+1.
BB (input/output) COMPLEX*16 array, dimension (LDBB, N)
On entry, the upper or lower triangle of the Hermitian band matrix B, stored in
the first kb+1 rows of the array. The j-th column of B is stored in the j-th col-
umn of the array BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for
max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
On exit, the factor S from the split Cholesky factorization B = S**H*S, as
returned by ZPBSTF.
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >= KB+1.
Q (output) COMPLEX*16 array, dimension (LDQ, N)
If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to
standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If
JOBZ = 'N', the array Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ
>= max(1,N).
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION If RANGE='V', the lower and upper bounds of the
interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A'
or 'I'.
IL (input) INTEGER
IU (input) INTEGER If RANGE='I', the indices (in ascending order) of the
smallest and largest eigenvalues to be returned. 1 <= IL <= IU <= N, if N > 0; IL
= 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
ABSTOL (input) DOUBLE PRECISION
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is
accepted as converged when it is determined to lie in an interval [a,b] of width
less than or equal to
ABSTOL + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If ABSTOL is less than or equal to zero, then
EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal
matrix obtained by reducing AP to tridiagonal form.
Eigenvalues will be computed most accurately when ABSTOL is set to twice the
underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0,
indicating that some eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('S').
M (output) INTEGER
The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and
if RANGE = 'I', M = IU-IL+1.
W (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) COMPLEX*16 array, dimension (LDZ, N)
If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the
i-th column of Z holding the eigenvector associated with W(i). The eigenvectors
are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.
WORK (workspace) COMPLEX*16 array, dimension (N)
RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
IWORK (workspace) INTEGER array, dimension (5*N)
IFAIL (output) INTEGER array, dimension (N)
If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO
> 0, then IFAIL contains the indices of the eigenvectors that failed to converge.
If JOBZ = 'N', then IFAIL is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is:
<= N: then i eigenvectors failed to converge. Their indices are stored in array
IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF
returned INFO = i: B is not positive definite. The factorization of B could not
be completed and no eigenvalues or eigenvectors were computed.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
LAPACK version 3.0 15 June 2000 ZHBGVX(l)